An interesting feature of gas-dominated source environments is that they are able to achieve a good fit to Auger observations even for spectral indices as soft as \(\gamma _\mathrm {inj} \sim -2\) , as is predicted for diffusive shock acceleration; see Fig. REF , which shows the best-fit accelerator spectral index for CRs. This is due to the fact that the hadronic interaction time depends very weakly on energy (see Fig. REF ), so that it is well approximated by a single power law \(\tau _g \sim E^\zeta \)  with \(\zeta \sim 0\) . Following the analysis of [1]} (see Section 2), this implies that quasi-ballistic-regime CRs (i.e. \(\tau _\mathrm {esc} \sim E^{-2}\) ) interacting dominantly with gas, can produce a spectrum escaping the source environment with hard spectral index \(\gamma _\mathrm {esc} \gtrsim -1\)  even when the injected spectral index is \(\gamma _\mathrm {inj} \sim -2\)  to \(-3\) .This mechanism can also be achieved by photohadronic interactions, since at high energies \(\tau _\gamma \sim E^0\)  as well. But in order for the peak of the CR spectrum to be in this regime the photon field needs a very high temperature and CRs require a large number of interactions on average. These two requirements cause the description of the UHECR spectrum and composition to degrade, as well as producing an excess of EHE neutrinos which violates the IceCube bounds. This clearly demonstrates that processing of the CR spectrum by the source environment can substantially modify the escaping spectral index to be as hard as is required by the Auger spectrum [2]}, without relying on exotic acceleration mechanisms.
